3.1.57 \(\int \frac {1}{(a+b (F^{g (e+f x)})^n)^2 (c+d x)^2} \, dx\) [57]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^2 (c+d x)^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*(F^(f*g*x+e*g))^n)^2/(d*x+c)^2,x)

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Rubi [A]
time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2),x]

[Out]

Defer[Int][1/((a + b*(F^(e*g + f*g*x))^n)^2*(c + d*x)^2), x]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2} \, dx &=\int \frac {1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^2 (c+d x)^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2),x]

[Out]

Integrate[1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2), x]

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Maple [A]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2} \left (d x +c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^2,x)

[Out]

int(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^2,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

1/(a^2*d^2*f*g*n*x^2*log(F) + 2*a^2*c*d*f*g*n*x*log(F) + a^2*c^2*f*g*n*log(F) + (F^(g*n*e)*a*b*d^2*f*g*n*x^2*l
og(F) + 2*F^(g*n*e)*a*b*c*d*f*g*n*x*log(F) + F^(g*n*e)*a*b*c^2*f*g*n*log(F))*F^(f*g*n*x)) + integrate((d*f*g*n
*x*log(F) + c*f*g*n*log(F) + 2*d)/(a^2*d^3*f*g*n*x^3*log(F) + 3*a^2*c*d^2*f*g*n*x^2*log(F) + 3*a^2*c^2*d*f*g*n
*x*log(F) + a^2*c^3*f*g*n*log(F) + (F^(g*n*e)*a*b*d^3*f*g*n*x^3*log(F) + 3*F^(g*n*e)*a*b*c*d^2*f*g*n*x^2*log(F
) + 3*F^(g*n*e)*a*b*c^2*d*f*g*n*x*log(F) + F^(g*n*e)*a*b*c^3*f*g*n*log(F))*F^(f*g*n*x)), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*(F^(f*g*x + g*e))^(2*n
) + 2*(a*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*(F^(f*g*x + g*e))^n), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{a^{2} c^{2} f g n \log {\left (F \right )} + 2 a^{2} c d f g n x \log {\left (F \right )} + a^{2} d^{2} f g n x^{2} \log {\left (F \right )} + \left (a b c^{2} f g n \log {\left (F \right )} + 2 a b c d f g n x \log {\left (F \right )} + a b d^{2} f g n x^{2} \log {\left (F \right )}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}} + \frac {\int \frac {2 d}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {c f g n \log {\left (F \right )}}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx + \int \frac {d f g n x \log {\left (F \right )}}{a c^{3} + 3 a c^{2} d x + 3 a c d^{2} x^{2} + a d^{3} x^{3} + b c^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c^{2} d x e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + 3 b c d^{2} x^{2} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}} + b d^{3} x^{3} e^{e g n \log {\left (F \right )}} e^{f g n x \log {\left (F \right )}}}\, dx}{a f g n \log {\left (F \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(F**(g*(f*x+e)))**n)**2/(d*x+c)**2,x)

[Out]

1/(a**2*c**2*f*g*n*log(F) + 2*a**2*c*d*f*g*n*x*log(F) + a**2*d**2*f*g*n*x**2*log(F) + (a*b*c**2*f*g*n*log(F) +
2*a*b*c*d*f*g*n*x*log(F) + a*b*d**2*f*g*n*x**2*log(F))*(F**(g*(e + f*x)))**n) + (Integral(2*d/(a*c**3 + 3*a*c
**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c**2*d*x*exp(e*g*
n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c*d**2*x**2*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + b*d**3*x**3*exp(e*g*n*
log(F))*exp(f*g*n*x*log(F))), x) + Integral(c*f*g*n*log(F)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x
**3 + b*c**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c**2*d*x*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*
c*d**2*x**2*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + b*d**3*x**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + I
ntegral(d*f*g*n*x*log(F)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b*c**3*exp(e*g*n*log(F))*exp
(f*g*n*x*log(F)) + 3*b*c**2*d*x*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)) + 3*b*c*d**2*x**2*exp(e*g*n*log(F))*exp(
f*g*n*x*log(F)) + b*d**3*x**3*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x))/(a*f*g*n*log(F))

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(1/(((F^((f*x + e)*g))^n*b + a)^2*(d*x + c)^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2),x)

[Out]

int(1/((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2), x)

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